We discuss how to apply many-body methods to correlated nanoscopic systems, and provide general criteria of validity for a treatment at the dynamical mean field theory (DMFT) approximation level, in which local correlations are taken into account, while nonlocal ones are neglected. In this respect, we consider one of the most difficult cases for DMFT, namely, for a quasi-one-dimensional molecule such as a benzene ring. The comparison against a numerically exact solution shows that nonlocal spatial correlations are relevant only in the limit of weak coupling between the molecule and the metallic leads and of low inter-atomic connectivity, otherwise DMFT provides a quantitative description of the system. As an application we investigate the role of correlations on electronic transport in quantum junctions, and we show that a local Mott-Hubbard crossover is a robust phenomenon in sharp nanoscopic contacts.
We discuss how to apply many-body methods to correlated nanoscopic systems, and provide general criteria of validity for a treatment at the dynamical mean field theory (DMFT) approximation level, in which local correlations are taken into account, while nonlocal ones are neglected. In this respect, we consider one of the most difficult cases for DMFT, namely, for a quasi-one-dimensional molecule such as a benzene ring. The comparison against a numerically exact solution shows that nonlocal spatial correlations are relevant only in the limit of weak coupling between the molecule and the metallic leads and of low inter-atomic connectivity, otherwise DMFT provides a quantitative description of the system. As an application we investigate the role of correlations on electronic transport in quantum junctions, and we show that a local Mott-Hubbard crossover is a robust phenomenon in sharp nanoscopic contacts.
Nanostructures with open shell transition metal or molecular constituents host often strong electronic correlations and are highly sensitive to atomistic material details. This tutorial review discusses method developments and applications of theoretical approaches for the realistic description of the electronic and magnetic properties of nanostructures with correlated electrons. First, the implementation of a flexible interface between density functional theory and a variant of dynamical mean field theory (DMFT) highly suitable for the simulation of complex correlated structures is explained and illustrated. On the DMFT side, this interface is largely based on recent developments of quantum Monte Carlo and exact diagonalization techniques allowing for efficient descriptions of general four fermion Coulomb interactions, reduced symmetries and spin-orbit coupling, which are explained here. With the examples of the Cr (001) surfaces, magnetic adatoms, and molecular systems it is shown how the interplay of Hubbard U and Hund’s J determines charge and spin fluctuations and how these interactions drive different sorts of correlation effects in nanosystems. Non-local interactions and correlations present a particular challenge for the theory of low dimensional systems. We present our method developments addressing these two challenges, i.e., advancements of the dynamical vertex approximation and a combination of the constrained random phase approximation with continuum medium theories. We demonstrate how non-local interaction and correlation phenomena are controlled not only by dimensionality but also by coupling to the environment which is typically important for determining the physics of nanosystems.
In strongly correlated materials, quantum fluctuations taking place at the level of atomic sites play a central role. These local processes are captured nonperturbatively in dynamical mean field theory (DMFT). Nonlocal mechanisms, however, are also suspected to be essential for the proper description of some of these materials, including the well-known high-temperature cuprate superconductors and the more exotic systems of atoms adsorbed on semiconducting surfaces.In this thesis, the effect of nonlocal, potentially long-range fluctuations beyond the DMFT framework is investigated from two angles.First, long-ranged interactions in adatom systems, computed from first principles within the constrained random-phase approximation (cRPA), are treated within the extended DMFT (EDMFT) framework and its combination with the diagrammatic GW method (GW +EDMFT). They are shown to explain some material trends observed in these compounds. In addition, thanks to the spatial and temporal information delivered by GW+EDMFT, they solve existing puzzles raised by different experimental conclusions drawn from different probes.Second, a new approach to the Hubbard model including nonlocal correlation effects is proposed. Formally based on the local expansion of the triply-irreducible functional, this approach, dubbed TRILEX, approximates the renormalized three-leg electron-boson vertex by a local but frequency-dependent vertex computed from an impurity model with retarded density-density and spin-spin interactions. This local vertex is used to construct frequency- and momentum-dependent self-energies capturing nonlocal correlation effects. By construction, this method reconciles two theoretical pictures of cuprate superconductors, namely the spin fluctuation theory at weak interaction strengths and the doped-Mott-insulator picture at strong coupling.This novel approach is applied to the single-band Hubbard model. It is shown to reduce to DMFT at strong coupling and high temperatures, and to fluctuation-exchange theories at weak coupling, while substantial deviations from both limits are observed at intermediate coupling. There, upon doping, antiferromagnetic spin fluctuations are observed to lead to a momentum-differentiated spectral intensity on the Fermi surface, which is reminiscent of the formation of Fermi arcs in photoemission measurements on cuprate superconductors.These two methodological developments are intrinsically linked to algorithmic aspects which are reviewed in the last part of this work. In particular, an open-source library dedicated to massively parallel computations in the field of quantum many-body physics, TRIQS, is presented, with a focus on error estimation through the binning and jackknife methods. The hybridization-expansion continuous-time quantum Monte-Carlo algorithm in the segment picture -- central to the TRILEX method -- and its state-of-the-art implementation are explained. Finally, as a further illustration, a new lightweight method relying on this algorithm and designed for realistic systems, the combined screened-exchange and dynamical DMFT method (SEx+DDMFT), is presented together with an application to the BaCo2As2 compound.
In this paper, we investigate the impact of nonlocal correlations on charge fluctuations in the two-dimensional single-band Hubbard model close to the Mott metal-to-insulator transition, employing the ladder dynamical vertex approximation. At half filling and for interaction strengths and temperatures where the system is in the Mott insulating phase, charge fluctuations are strongly suppressed. Under these conditions, dynamical mean-field theory (DMFT) calculations predict a strong enhancement of the charge susceptibility at small (electron or hole) doping. However, these DMFT results include only the effects of purely local correlations despite the importance of nonlocal correlations in two-dimensional systems. We have, hence, carried out ladder dynamical vertex approximation $(\mathrm{lD}\mathrm{\ensuremath{\Gamma}}\mathrm{A})$ simulations which allow for the inclusion of such nonlocal correlation effects while retaining the local ones of DMFT. Our $\mathrm{lD}\mathrm{\ensuremath{\Gamma}}\mathrm{A}$ numerical data show that close to half filling the large uniform charge susceptibility of DMFT is strongly suppressed by nonlocal fluctuations but gradually increases with (electron) doping. At a certain doping value, charge fluctuations eventually become larger in $\mathrm{lD}\mathrm{\ensuremath{\Gamma}}\mathrm{A}$ with respect to DMFT, indicating that the absence of nonlocal correlations underestimates the mobility of the charge carriers in this parameter regime. This metallization effect is also reflected in an enhancement of the $\mathrm{lD}\mathrm{\ensuremath{\Gamma}}\mathrm{A}$ kinetic and potential energies and a corresponding reduction of the (absolute value of the) $\mathrm{lD}\mathrm{\ensuremath{\Gamma}}\mathrm{A}$ Matsubara self-energy with respect to DMFT.
Recent developments in methods and computational power render it possible to realistically simulate nanoscopic systems such as surfaces, two-dimensional materials, and nanodots including strong electronic correlations. Nanoscopic structuring enables the tailoring of the electronic structure which can be the basis of future electronic devices. This thesis addresses method developments and applications at the interface of ab-initio methods and model based many-body methods for the case of nanoscopically structured systems with strong correlations. In contrast to bulk materials, low-dimensional materials exhibit long-range interactions due to reduced screening. In this work, the general question how these long-range interactions affect electronic properties is investigated. To this end, a variational approach which approximates models with long-range interactions by models with only local interactions is introduced. For the case of an ab-initio derived model of graphene it is found that nonlocal interactions stabilize the semimetallic phase. The quality of this approach is discussed using a simple test case. Realistic models of interacting impurities embedded in an extended solid involve a large amount of bath sites and low symmetries regarding the impurity, which renders an exact treatment impossible. A variational algorithm is presented which optimizes corresponding exactly solvable effective models. Thus, the method is a proposition to an unambiguous solution of the so-called bath-discretization problem in exact diagonalization approaches to the Anderson impurity model. The method is benchmarked for a simple test case and applied to realistic models of Co atoms in Cu hosts and Fe atoms on alkali surfaces. Finally, the (001) surface of Cr is investigated by incorporating local correlation effects into a material realistic description derived from density functional theory. To this end, the LDA DMFT method is used to calculate spectral functions, which are compared to spectroscopic experimental data. So far open experimental features are thereby clarified. Cr(001) exemplifies a situation where correlation effects are determined by the geometric structure of a material: While correlations effects are weak in bulk Cr, they are key for the electronic structure of the Cr(001) surface.
Strong electronic correlations are the root of many fascinating phenomena of condensed matter physics. However, they also make calculations notoriously difficult. This thesis is concerned with the development and improvement of computational methods for strongly correlated electron systems. After a brief recapitulation of the Green’s function formalism,which is the theoretical basis of most such methods, we discuss the analytical continuation from imaginary to real frequencies. Especially the maximum entropy method is explained in detail and an implementation there of is presented.The remainder of this thesis deals with methods to compute Green’s functions on thei maginary frequency axis. First, the Anderson impurity model (AIM) is considered. It can be solved numerically – a task for which several methods exist. We focus on continuous-time quantum Monte Carlo with worm sampling, through which various kinds of correlation functions can be calculated. On this basis we derive and implement symmetric improved estimators that allow for the computation of the self-energy and vertex functions where the level of stochastic noise is largely reduced with respect to conventional estimators.As a next step the dynamical mean-field theory (DMFT) is introduced as a method to capture local correlations in the Hubbard model (HM) by self-consistently mapping it onto the AIM. We study the convergence behavior of this iterative procedure and the improvement of convergence due to mixing. Then, we compare electron scattering rates from DMFT in the weak-coupling regime of the square-lattice HM to results from Boltzmann scattering theory.While local correlations account for many interesting phenomena, such as the Mott-Hubbard metal-insulator transition, they are not sufficient to correctly describe, e.g., the physics of phase transitions in two- or three-dimensional systems. It is thus the final step in this thesis to also include nonlocal correlations by means of the dynamical vertex approximation (DΓA). For this we develop a new self-consistency scheme, which is then applied to several systems: We study the square-lattice HM, where we compare to benchmark methods in the weak-coupling regime, but also present results for intermediate and strong coupling. Furthermore, a two-band model with doping and a three-band Wannier model for strontium vanadate are studied. As an application of our new self-consistent DΓA, we study the effects of geometrical frustration in the kagome lattice, where we also compare our results to determinant quantum Monte Carlo.
Motivated by the intriguing physics of quasi-two-dimensional fermionic systems, such as high-temperature superconducting oxides, layered transition metal chalcogenides, or surface or interface systems, the development of many-body computational methods geared at including both local and nonlocal electronic correlations has become a rapidly evolving field. It has been realized, however, that the success of such methods can be hampered by the emergence of noncausal features in the effective or observable quantities involved. Here, we present an approach wherein local many-body techniques such as dynamical mean-field theory (DMFT) are extended to nonlocal correlations and interactions, which preserves causality and has a physically intuitive interpretation. Our strategy has implications for the general class of DMFT-inspired many-body methods and can be adapted to cluster, dual boson, or dual fermion techniques with minimal effort.